Model for a Super-High-Tc Superconductor
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Problem of Super-High-Tc -Superconductivity
By Alexei A. Abrikosov Materials Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439
Abstract
A possibility of a general path for the search of “Super-High-Tc” (SHT) materials is discussed. The approach is based on the idea of enhancement of the basic parameters, entering the BCS formula for Tc : the electron-phonon interaction constant, the density of states and the energy of mediating quasiparticles. The known examples of materials with a relatively high Tc are examined. The author’s idea of finding a substance forming a “crystal1ine excitonium” phase (1978) is presented. Since the BCS formula represents only an isotropic model, the role of anisotropy is analyzed within the author’s model for high-Tc layered cuprates. The main conclusion is, that anisotropy of the spectrum and a long ranged interaction widens the options for obtaining SHT.
The submitted manuscript has been created by the University of Chicago as Operator of Argonne National Laboratory (“Argonne”) under Contract No. W-31-109-ENG-38 with the U.S. Department of Energy. The U.S. Government retains for itself, and others acting on its behalf, a paid-up, non-exclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government.
PACS - 74.20.Fg, 74.25.Bt, 2
At present it is well established that superconductivity is a consequence of formation of Cooper pairs. The latter takes place, if the attractive forces between electrons exceed the Coulomb repulsion. In spite of many other suggestions, only two mechanisms of such an attraction are definitely established: namely, the exchange of phonons and of spin-fluctuations. The latter mechanism is the cause of superconductivity in some heavy-fermion metals, e.g., UPt3, and of superfluidity of liquid He3. This can be concluded from the unusual form of the order parameter, resulting in power-law temperature dependencies of the specific heat and the thermal conductivity at low temperatures. For He3. an estimate based on the assumption of a Van der Waals attraction mechanism led to a much lower estimate for the critical temperature (~10-6 K instead of ~10-3 K). Experiment shows that the critical temperature in all such cases is low, not higher, than 1 K. This may be due to the fact, that the exchange interaction, on which Tc depends exponentially, is always smaller, than the main Coulomb interactions. About the phonon-mediated interaction there existed a belief, that it cannot exceed some limiting value. This idea was based on a formula, derived by Migdal in the same paper [1], as the famous “Migdal’s theorem”. It was a connection between the “bare” phonon frequency, which was due to the elasticity of the crystalline lattice, and the “renormalized” frequency, which took into account the electron-phonon interaction. At small wave numbers it was
k = 0 k 1 2 , where g 2 , g - the electron-phonon interaction and - the density of states. If exceeded ½ , k became imaginary, and this meant the instability of the crystalline lattice. Since T c depended on , this lead to a limitation: T c < 40 K. At first it seemed that this limitation is true but even before high- T c superconductors were discovered, serious objections appeared. First of all, Migdal’s derivation was based on the assumption 1, which was not valid for 1/2 . The second objection was that 0 k was not a physical quantity, and therefore, its connection with the measurable frequency had no physical
meaning. From this it followed that much higher values of T c , than 40 K, could be expected. A special case presented the high-Tc layered cuprates. Unusually high values of Tc and the absence of the isotope effect in some of them led to the conclusion that the mechanism of Cooper pairing in these substances is different from phonons. Several options were suggested: spin fluctuations, resonating valence bonds (RVB) with “pair tunneling”, Coulomb repulsion at the copper sites, et al. All these ideas eventually collapsed, either as 3
contradicting experiment (RVB), or on purely theoretical grounds (e.g., repulsion alone can exclude some possibilities but cannot create new ones). On the other hand, there were definitive experiments stressing the importance of phonons: large isotope effect in substituted and underdoped substances, clear similarity of the tunneling conductance at high voltages and the phonon density of states, as in conventional low-Tc materials, phonon-assisted Josephson effect and change of the slope of the electron linear energy spectrum in the normal state at the phonon energy. In the discussion about the mechanism of high-Tc superconductivity in cuprates plenty of passion was involved, which led to suppression of references to “unpleasant” experimental data, referring to all fashionable, even mutually contradicting, theories and other violations of scientific ethics. This resulted in the appearance of a legend about an “unsolvable mystery of the high-Tc superconductivity”. In reality, a theory exists [2], based not on abstract models but on the peculiar properties of the electron energy spectrum in real substances, first of all, the quasi-two-dimensionality of electrons in the CuO2 planes and the “extended saddle point singularities” along the Brillouin zone boundaries, found in many substances and confirmed by computation. In addition, on the basis of many experiments, a hypothesis of the resonant tunneling connection between the CuO2 planes was proposed. This theory explained most of the unusual properties of the layered cuprates but it also predicted, that the possibility of a further increase of the critical temperature in such substances is unlikely. Therefore, a search for a different model, focused on increasing Tc, has to be started. From the existing experience with new types of superconductors with an elevated Tc it can be concluded, that the famous BCS formula (see, e.g., [3]) 2 Tc E 0 exp1/g ,
( E 0 is the energy of quasiparticles, mediating the interaction between electrons, g is the the electron – quasiparticle interaction constant, and - the density of states of the electrons) explains very well the cause of their relatively high Tc. For doped fullerenes it is an enhanced E 0 , which is the energy of radual vibrations of carbon atoms in the bucky-balls C60, as well as a large effective mass (rare hoppong between the bucky-balls), and hence, a large . For MgB2 – a large value of g. For layered cuprates – a large due to quasi-one-dimensionality of the spectrum near the extended saddle point singularities. The goal would be to increase at least one of the three parameters, entering the BCS formula. 4
Although, this was exactly what happened in the examples given above, the discovery of these substances was due to chance, except for the layered cuprates, where their discoverer, Alex Mueller was guided by the idea of in- creasing the interaction constant g in perovskytes, due to the Jahn-Teller effect. This is doubtful, but “victorious generals cannot be tried in court”, as the Russian Empress, Katherine the Great, once said. Invention of a regular leading idea for finding room-temperature superconductors requires, first of all, understanding which of the three important parameters can be tuned. The most difficult is the tuning of the interaction. Mueller’s idea about finding a substance with a Jahn-Teller enhancement of the electron-phonon interaction eventually failed. There is a rule, that “bad conductors make good superconductors”, and it reflects the fact that the increase of the electron-phonon interaction leads to higher Tc and also to a higher resistance in the normal state but this is far from really high Tc’s. The second idea could be the enhancement of . This is not so hopeless. On the basis of the examples, given above, one can think about the increase of the effective mass, or about reduced dimensionality (layered, or quasi- one-dimensional, materials). The attempt to use the first of these possibilities is undermined by the uniqueness of fullerenes. Nevertheless, one can think about compounds with a good attachment of electrons to certain atoms, or layers, with long distances between them. What concerns the reduced dimensionality; the existing substances of this kind (mostly organics) have rather low critical temperatures, and provide little inspiration. The appearance of extended saddle point singularities (flat bands) has no qualitative explanation, although, as we already mentioned, it was confirmed by the LDA computations of the band structure. The simplest to imagine is the increase of the characteristic energy of the mediating quasiparticles, since it defines the scale E 0 in the BCS formula. This was the basis of the suggestions by W. Little [4], V. Ginzburg [5], D. Allender and J. Bardeen [6] et al., where the main point was to replace the phonons by excitons, having much larger excitation energies. Unfortunately, these ideas did not provide high Tc’s, most probably, because the interaction of electrons with the corresponding excitons was week. In principle, the role of such excitons could be played by excitations of electrons in the inner shells of transitional metal atoms but there was a restriction on the degree of hybridization with the outer shells, which was either violated, or decreased drastically the interaction. The idea, which we would like to propose [7], belongs formally to the “excitonic” class, but hopefully, does not diminish the interaction strength. 5
In the 70-ies different special features of metal-insulator transitions were discussed. Among them was the idea if formation of the “crystalline excitonium”, proposed first by C. Herring and described in the paper by B. Halperin and T. M. Rice [8]. It was suggested that in an “even” metal (or semimetal) with equal numbers of electrons and holes, located at different pockets of the Brillouin zone, in the case, if the ratio of effective masses is very high, the heavy carriers (say, holes) can form a “Wigner crystal” through which the other carriers move, as a liquid. The whole structure forms an artificial metal, which can be superconducting. The advantage, compared to conventional metals, is that the heavy holes are, nevertheless, thousands of times lighter, than the ions in conventional metals, and,
correspondingly the scale of corresponding “phonons”, E 0 , can be drastically increased without a loss in the strength of the interaction. This idea was proposed in the 70-ies, as a possible explanation of hints of high-temperature superconductivity in copper chloride. Since this superconductivity was never definitely confirmed, the theory was not developed further, and then the discovery of high- Tc layered cuprates switched all the attention to them. The development of this idea must start with a more reliable derivation of the critical carrier mass ratio. The next step must be the search of a possible object, for which the experience of band-structure specialists would be most valuable. If all this succeeds, comes the turn of crystallographers for actual synthesis of such metals. All this reasoning was based on the BCS formula, which was derived for an isotropic model with a short-range interaction. More general assumptions widen enormously the variety of possibilities. I will give you only one example, which refers to a realistic case [9]. In our model of layered cuprates the main role is played by the “extended saddle point singularities”, where the electron motion becomes one- dimensional, and in order to explain the appearance of a d-wave order parameter, the electron-phonon interaction is assumed to be long-ranged due to bad Coulomb screening. Then, it can be shown, that Migdal’s theorem remains valid even in the case of a small chemical potential (calculated from the bottom of the band), provided that 2 2 m , where is the characteristic phonon frequency, and - the inverse Debye 2 T screening radius. In the case, if c 2 m , 8 2 d 2 T c e x p 2 , g m
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where =1.78, g is the electron-phonon interaction constant and d -the 2 interlayer distance. In the opposite case, /2m Tc , 2 2 g 2 m T c 0 . 5 7 2 . 2 m 2 d From these formulas it follows that anisotropy and the long-range interaction
can enhance the T c quite considerably.
Acknowledgment
I would like to thank the Organizers of the “Workshop on the possibility of room temperature superconductivity and related topics” at the University of Notre Dame. This work was supported by the Department of Energy under the contracts # W-31-109-ENG-38.
References
[1] A. B. Migdal, Soviet Physics, JETP 5 (1957) 333 [2] A. A. Abrikosov, International Journal of Modern Physics B 13 (1999) 3405; Physica C 341-348 (2000) 97 [3] A. A. Abrikosov, ‘Fundamentals of the theory of metals”, Elsevier Science Publishers, 1988 [4] W. A. Little, Phys. Rev. A 134 (1964) 1416 [5] V. L. Ginzburg, Zh. Eksp. & Teor. Fiz. 47 (1964) 2318 [6] D. Allender et al., Phys. Rev. 7 (1973) 1020; ibid. 8 (1973) 4433 [7] A. A. Abrikosov, JETP Lett. 27 (1978) 219 [8] B. I. Halperin & T. M. Rice, Rev. Mod. Phys. 40 (1968) 755 [9] A. A. Abrikosov, Physica C, in press